Acoustic wake in an isothermal profile: dynamical friction and gravitational wave emission

TL;DR

This paper models the acoustic wake and dynamical friction of a perturber in an isothermal sphere, revealing conditions that maximize gravitational wave emission, with implications for black hole inspirals.

Contribution

It provides an analytical solution for the acoustic wake and dynamical friction in a singular isothermal sphere, highlighting resonance effects that enhance gravitational wave emission.

Findings

Dynamical friction is suppressed at subsonic speeds in an isothermal sphere.

Resonance near the constant circular velocity maximizes GW emission.

GW signals could be comparable to vacuum emission for black hole inspirals.

Abstract

We consider the motion of a circularly-moving perturber in a self-gravitating, collisional system with spherically symmetric density profile. We concentrate on the singular isothermal sphere which, despite its pathological features, admits a simple polarization function in linear response theory. This allows us to solve for the acoustic wake trailing the perturber and the resulting dynamical friction, in the limit where the self-gravity of the response can be ignored. In steady-state and for subsonic velocities $v_p<c_s$, the dynamical friction torque $F_\varphi\propto v_p^3$ is suppressed for perturbers orbiting in an isothermal sphere relative to the infinite, homogeneous medium expectation $F_\varphi\propto v_p$. For highly supersonic motions, both expectations agree and are consistent with a local approximation to the gravitational torque. At fixed resolution (a given Coulomb logarithm), the response of the system is maximal for Mach numbers near the constant circular velocity of the singular isothermal profile. This resonance maximizes the gravitational wave (GW) emission produced by the trailing acoustic wake. For an inspiral around a massive black hole of mass $10^6 M_\odot$ located at the center of a (truncated) isothermal sphere, this GW signal could be comparable to the vacuum GW emission of the black hole binary at sub-nanohertz frequencies when the small black hole enters the Bondi sphere of the massive one. The exact magnitude of this effect depends on departures from hydrostatic equilibrium and on the viscosity present in any realistic astrophysical fluid, which are not included in our simplified description.